We present scanning tunneling microscopy (STM) studies of superconductivity of single layer Pb and Pb islands on Si(111). We perform temperature-dependent measurements of the differential conductance from 0.38 to 4 K in fields of up to 6 T to extract the critical temperature T_C for the onset of superconductivity. We find T_C = 1.5 K for a single layer Pb on Si(111), and a critical out-of-plane field of 150 mT. This deviates from bulk Pb, where T_C = 7.2 K and H_C = 80 mT are reported. Our results provide the temperature dependence of the superconducting gap 2 ∆. A description of this dependence in the framework of the Bardeen-Cooper-Schrieffer theory indicates an energy gap of 2 ∆₀ = 0.7 meV, considerably less than the 2.7 meV found for bulk Pb. We observe that the insertion of a single layer Ag between Pb and Si suppresses superconductivity in the Pb film even at the lowest temperature of 0.38 K. Pb islands on Ag/Si(111) exhibit superconductivity. Our position-dependent STM studies from a nine-layer tall Pb island into the surrounding non-superconducting Pb wetting layer on Ag/Si(111) reveal an extension of a superconducting gap in spectroscopy up to 20 nm away from the Pb island edge at 0.38 K.